[论文解读] Enumerating Markov Equivalence Classes of Acyclic Digraph Models

Steven B. Gillispie, Perlman, Michael D.|arXiv (Cornell University)|Jan 10, 2013

Bayesian Modeling and Causal Inference参考文献 31被引用 73

一句话总结

本文利用Pearl和Verma的等价性准则，通过优化的有序算法对无环有向图（DAG）的马尔可夫等价类进行计算枚举，生成无向图并将DAG分类为等价类。主要发现为：等价类与DAG的数量之比随顶点数增加渐近趋近于约0.267，类大小和边分布分别呈现高斯和不规则模式。

ABSTRACT

Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.

研究动机与目标

• 系统枚举无环有向模型的马尔可夫等价类，以改善模型设定与搜索效率。
• 分析顶点数不超过10的DAG中，按边数和类大小划分的等价类分布。
• 开发并应用一种有序算法的新变体，用于生成无向图，作为等价类枚举的基础。
• 确定任意单一无向图能生成的最大等价类数量，观察其阶乘增长特性。

提出的方法

• 基于Pearl和Verma的马尔可夫等价性准则实现计算机程序，将DAG分类为等价类。
• 对有序算法进行改进，用于生成无向图，并引入一种新颖变体以支持高效的等价类枚举。
• 枚举所有顶点数不超过10的DAG，随后利用等价性准则将其分组为马尔可夫等价类。
• 对等价类按边数和类大小进行统计分析，包括分布模式与渐近行为。
• 计算单个无向图能生成的最大等价类数量，发现其近似呈阶乘增长。

实验结果

研究问题

• RQ1 随着顶点数增加，马尔可夫等价类与DAG的数量之比的渐近值是多少？
• RQ2 DAG中按边数划分的等价类如何分布？
• RQ3 类大小的频率分布如何？是否存在可预测的模式？
• RQ4 单个无向图能生成的最大等价类数量是多少？其随图大小如何变化？
• RQ5 新版本的有序算法在生成等价类时，其性能与结构与标准方法相比有何差异？

主要发现

• 随着顶点数增加，马尔可夫等价类数量与DAG数量之比渐近趋近于约0.267。
• 按边数划分的等价类分布趋近于高斯（正态）分布，表明各类中边数具有集中趋势。
• 大小为1的类最为常见，较大类的占比先下降，随后呈现更不规则的模式。
• 任意单一无向图生成的最大等价类数量随顶点数增加近似呈阶乘增长。
• 新版本的有序算法成功支持了等价类的枚举，并实现了顶点数达10的可扩展计算。